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# 数学代写|凸优化代写Convex Optimization代考|Main lemma

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## 数学代写|凸优化代写Convex Optimization代考|Main lemma

At this moment we are interested in the following problem:
$$\min {f(x) \mid x \in Q}$$
where $Q$ is a closed convex set, and $f$ is a function, which is convex on $R^n$. We are going to study some methods for solving (3.2.3), which employ subgradients $g(x)$ of the objective function. As compared with the smooth problem, our goal now is much more complicated. Indeed, even in the simplest situation, when $Q \equiv R^n$, the subgradient seems to be a poor replacement for the gradient of smooth function. For example, we cannot be sure that the value of the objective function is decreasing in the direction $-g(x)$. We cannot expect that $g(x) \rightarrow 0$ as $x$ approaches a solution of our problem, etc.

Fortunately, there is one property of subgradients that makes our goals reachable. We have proved this property in Corollary 3.1.4:
At any $x \in Q$ the following inequality holds:
$$\left\langle g(x), x-x^*\right\rangle \geq 0$$
This simple inequality leads to two consequences, which form a basis for any nonsmooth minimization method. Namely:

• The distance between $x$ and $x^*$ is decreasing in the direction $-g(x)$.
• Inequality (3.2.4) cuts $R^n$ on two half-spaces. Only one of them contains $x^*$.

Now we are ready to analyze the behavior of some minimization schemes. Consider the problem
$$\min {f(x) \mid x \in Q}$$
where $f$ is a convex on $R^n$ function and $Q$ is a simple closed convex set. The term “simple” means that we can solve explicitly some simple minimization problems over $Q$. In accordance to the goals of this section, we have to be able to find in a reasonably cheap way a Euclidean projection of any point onto $Q$.

We assume that problem (3.2.7) is equipped with a first-order oracle, which at any test point $\bar{x}$ provides us with the value of objective function $f(\bar{x})$ and with one of its subgradients $g(\bar{x})$.

As usual, we try first a version of a gradient method. Note that for nonsmooth problems the norm of the subgradient, $|g(x)|$, is not very informative. Therefore in the subgradient scheme we use a normalized direction $g(\bar{x}) /|g(\bar{x})|$.

1. Choose $x_0 \in Q$ and a sequence $\left{h_k\right}_{k=0}^{\infty}$ :
$$h_k>0, \quad h_k \rightarrow 0, \quad \sum_{k=0}^{\infty} h_k=\infty .$$
2. $k$ th iteration $(k \geq 0)$.
Compute $f\left(x_k\right), g\left(x_k\right)$ and set
$$x_{k+1}=\pi_Q\left(x_k-h_k \frac{g\left(x_k\right)}{\left|g\left(x_k\right)\right|}\right) .$$

## 数学代写|凸优化代写Convex Optimization代考|Main lemma

$$\min {f(x) \mid x \in Q}$$

$$\left\langle g(x), x-x^*\right\rangle \geq 0$$

$x$和$x^*$之间的距离沿$-g(x)$方向减小。

$$\min {f(x) \mid x \in Q}$$

$k$ 迭代$(k \geq 0)$。

$$x_{k+1}=\pi_Q\left(x_k-h_k \frac{g\left(x_k\right)}{\left|g\left(x_k\right)\right|}\right) .$$

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